# Euler's pioneering equation

Since The Mathematical Intelligencer conducted a poll in 1988 about which was the most beautiful among twenty-four theorems. Euler's equation $e^{i\pi}+1=0$ or $e^{iπ}=−1$ turned out to be the winner, and that is still today largely accepted among mathematicians. Even among physicists this is true. In a similar poll from 2004, it came out second after Maxwell's equations. The subtitle of this book is therefore *The most beautiful theorem in mathematics*.

This may immediately raise some controversy, not about the choice of the formula, but perhaps about what it should be called: a theorem, an identity, an equality, a formula, an equation,... A theorem or a formula applies but these are quite general terms. The others refer to formulas with an equal sign. The term identity assumes that there is a variable involved and that the formula holds whatever the value of that variable. That applies to Euler's identity, which is the related formula $e^{ix}=\cos(x)+i\sin(x)$. The previous formula appears as a special case of this identity. Wilson calls the former formula and "equation" but the reader with some affinity to the French language would probably prefer to call it an equality because the French équation means it has to be solved for an unknown variable. But all the previous names have been used interchangeably to indicate the formula. Calling it *Euler's identity* may not be the most correct but it is probably the most common terminology.

Whatever it is called, the description, if not *most beautiful*, then certainly the qualification *most important* or *most remarkable*, would be well deserved. It involves five fundamental mathematical constants: 1,0,π,e, and i in one simple relation. The 1 generates the counting numbers. The zero took a while to be accepted as a number but also negative numbers were initially considered to be exotic. Rational numbers were showing up naturally in computations, but so did numbers like √2 and π. These required an extension of the rationals with algebraic irrationals like √2 and the transcendentals like π which results in the reals that include all of them. The constant e (notation by Euler) relates to logarithms and its inverse the exponential function growing faster than any polynomial. Finally the imaginary constant i = √-1 (which is another notation introduced by Euler) was needed to solve any quadratic equation. This i allowed to introduce the complex numbers so that the fundamental theorem of algebra could be proved. The exponential and complex exponential are essential in applied mathematics. Euler's identity is most remarkable because it relates exponential growth or decay of the real exponential, and the oscillating behaviour of sines and cosines in the complex case.

All these links allow Wilson to tell many stories about mathematics that are usually discussed in books popularizing mathematics for the lay reader. There are indeed five chapters whose titles are the five previous constants and a sixth one is about Euler's equation. He does this in a concise way. The amount of information compressed in only 150 pages is amazing. This doesn't mean that it is so dense that it becomes unreadable. Quite the opposite. Because there are no long drawn-out detours, the story becomes straightforward and understandable. For example the first chapter (only 17 pages including illustrations) introduces children's counting rhymes, compares the names for numbers in seven different languages, and compares number systems: Roman, Egyptian, Mesopotamian, Greek, Chinese, Mayan, and the Hindu-Arabic. The latter was popularized in the West by Fibonacci and Pacioli. There are many illustrations not only of the notation of these different numerals in this chapter, but there are in fact many other illustrations throughout the book. This does not increase the number of pages needlessly because a picture sometimes says more than a thousand words. There are no colour illustrations but colour is not relevant for what they represent.

This is not the first book on Euler's equation. For example Paul Nahin. *Dr. Euler's Fabulous Formula*, Princeton University Press (2006), which is a bit more mathematically advanced, and a more recent one by David Stipp. *A Most Elegant Equation*, Basic Books (2017), which has more info about the person Euler. In the current book Euler's name appears frequently but as a person he is largely absent. For most of the five constants, separate popularizing books have been written or they are discussed in a chapter of more general popular books about mathematics, too many to list them here. Wilson refers to some of them in an appendix with a short list of additional literature, conveniently listed by subject.

There is of course mathematics in this book. It would be weird if there wasn't. But there is nothing that should shy away a reader with a slight affinity for mathematics. Some of it can be skipped, but the exponential and trigonometric functions, series, and an occasional integral do appear. The more advanced definitions or computations, are put in one of the eleven grey-shaded boxes distributed throughout the book, so that skipping is easy. Most of the topics are placed in their historical context. For example, the history of the computation of π is well represented, and also the history of the logarithms as they were derived by Napier and Briggs and how they relate is nicely explained. There are some notes to explain how complex numbers can be generalised to quaternions and even octonions, and several examples from applied mathematics illustrate the meaning and relevance of the exponential function.

A minor glitch: Albert Girard (1595-1632) who was the first to have formulated the fundamental theorem of algebra, is called on page 116 a Flemish mathematician, which is strange because the man was born in France, but, as a religious refugee, moved to Leiden in what was then the Dutch Republic of the Netherlands. So I do not think that the characterization Flemish does apply here.

The book does not go deep into the subjects discussed, but I liked it because it is quite broad, touching upon so many mathematical subjects, mainly in their historical context, while readability remains most enjoyable notwithstanding its conciseness.

**Submitted by Adhemar Bultheel |

**14 / Apr / 2019